V. Breazu-Tannen. Combining algebra and higher-order types. In Proc. 3rd IEEE Symposiu on Logic in Compu6B Science, Edinbu gh (UK), July 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....their expressive power, higher order logicsa re widely used for specifica tiona nd verifica tion. For the extension of term rewriting in this direction, there exist severa l di#erent forma lisms which integra te typed la mbda ca lculusa nd term rewrite systems, including Klop [13] Brea zu Ta nnen [3]a nd Nipkow [22] We follow thea pproa ch given in the la tter work, wherea rewriting rela tion modulo # , # a nd# conversion is considered. In thispa per wea da pt the first order proof method ca lled inductionles s induction, orproof by cons is tency , to the higher order setting. Thera tiona ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proc. 3rd IEEE Symposiu on Logic in Compu6B Science, Edinbu gh (UK), July 1988.

.... There existseveraldioeerentformalismsontheintegrationoftypedlambdacalculusandrewritesystems, andonthestudyoftheinteractionbetween braicrewriting and fi reduction.Theearliest oneseemstobetheworkon combinatory reduction systems duetoKlop[12] In[3]Breazu Tannencombinestherules for termswith arbitrary rst orderrewrite rules,andmodularproperties ofthecombinedsystem are proved.MorerecentlyJouannaudandOkada[10]introduced the AlgebraicFunctionalLanguage allowingtodenehigher orderfunctionalconstants byrewriterules ....

V.Breazu-Tannen. Combining algebraandhigher-ordertypes.In Proceedings3rd IEEESymposiumonLogicinComputerScience,Edinburgh(UK),July 1988.

....by pattern matching, provided all righthand side recursive calls are structurally smaller than the left hand side call [12] His notion is very abstract, though, and relies on a well foundedness assumption which is satisfied in practice. Concurrently, following the pioneering works of Tannen [8], Tannen and Gallier [9,10] and Okada [40] the last two authors of the present paper proposed another solution, for a polymorphically typed calculus, based on pattern matching functional definitions following the so called General Schema [27,28] This work was then generalized so as to cover ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proc. of the 3rd Symp. on Logic in Computer Science, IEEE Computer Society, 1988.

....rewriting [6] There exist several different formalisms on the integration of typed lambda calculus and rewrite systems, and on the study of the interaction between algebraic rewriting and fi reduction. The earliest one seems to be the work on combinatory reduction systems due to Klop [12] In [3] Breazu Tannen combines the rules for terms with arbitrary first order rewrite rules, and modular properties of the combined system is proved. More recently Jouannaud and Okada [10] introduced the Algebraic Functional Language allowing to define higher order functional constants by rewrite rules ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, Edinburgh (UK), July 1988.

....important assumption is that the reduction relation = R [ fi must be confluent. We will try to find sufficient conditions on R in order to get the confluence of R [ fi . In the simplytyped calculus, if R is a first order rewrite system then the confluence of R is a sufficient condition [7]. But few results are known in the case of a richer type system or of higher order rewriting. ffl Finally, we expect to extend this work with rewriting modulo some useful equational theories like associativity and commutativity, and also by allowing j reductions in the type conversion rule. 9 ....

V. Breazu-Tannen. Combining algebra and higherorder types. In Proc. of LICS'88, IEEE Computer Society.

....of logic programming languages and theorem provers like Prolog [21] or Isabelle [25] In particular, he extended to the higher order case the decidability result of Knuth and Bendix about local confluence of first order term rewrite systems. 1 At the same time, after the works of Breazu Tannen [6], Breazu Tannen and Gallier [7] and Okada [24] on the combination of Church s simply typed calculus with first order term rewriting, Jouannaud and Okada introduced higher order algebraic specification languages [11, 12] to provide a computational model for typed functional languages extended with ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proc. of LICS'88, IEEE Computer Society.

....notion of unification, a notion of principal typing which is more general than ML s principal type property, since also the types for the free variables of terms are inferred. Introduction Since the first investigations on combinations of Lambda Calculus (LC) and Term Rewriting Systems (TRS) [13, 19, 14, 26], this topic has drawn attention from the theoretical computer science community. At first, the area of programming language design was consider to be the typical field on which the theoretical results for combinations of the two computational paradigms could better exploit their potentialities. ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings of the Third Annual IEEE Symposium on Logic in Computer Science, pages 82--90, 1988.

.... Two Level Approach towards Lean Proof Checking Gilles Barthe # , Mark Ruys and Henk Barendregt May 15, 1996 Abstract We present a simple and e#ective methodology for equational reasoning in proof checkers. The method is based on a two level approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with ....

.... Two Level Approach towards Lean Proof Checking Gilles Barthe # , Mark Ruys and Henk Barendregt May 15, 1996 Abstract We present a simple and e#ective methodology for equational reasoning in proof checkers. The method is based on a two level approach distinguishing between syntax and semantics of mathematical theories. The method is very general and can be carried out in any type system with inductive ....

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V. Breazu-Tannen. Combining algebra and higher-order types, in the proceedings of LICS'88, pp 82-90, IEEE, 1988.

....algebraic features allowing one, in particular, to deal with equality in an ecient way. In the rst case, we nd the works on CRS [KvOvR93] XRS [Pag98] and other higher order rewriting systems [Wol93, NP98] in the second case the works on combination of calculus with term rewriting [Oka89, BT88, GBT89, JO97] to mention only a few. Our previous works on the control of term rewriting [KKV95, Vit94, BKKR01] led us to introduce the calculus. Indeed we realized that the tool that is needed in order to control rewriting should be made explicit and could be itself naturally described using ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, Edinburgh (UK), pages 82-90, 1988.

....expansions as the vertical one. Left linearity makes it easy to show that (DPG) holds also taking the algebraic system T as the horizontal reduction and expansions as the vertical reduction. Taken together, these two facts give F#T ### 2 #SP fflffl ### 2 #SP F#T We know from [6, 8] that combining the non extensional simply typedlambda calculus with a confluent first order algebraic rewriting system preserves confluence. On the other hand, this combination yields a strongly normalizing system when the algebraic one is [7, 25] This is enough to apply lemma 7 and obtain ....

V. Breazu-Tannen. Combining algebra and higher order types. In LICS, pages 82--90, July 1988.

....In order to design more expressive higher order logic programming systems enhanced with a first order equational theory E, one should consider higher order E unification, i.e. unification modulo = fij and =E . The problem of combining calculi with first order equational theories was initiated in [BT88] and the higherorder E unification problem has been already successfully studied [QW96, NQ91, Sny90] by extending the techniques developed [Hue75] for unification of simply typed terms. On the side of functional programming languages implementations, the operation of substitution (issued from ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings, Third Annual Symposium on Logic in Computer Science, pages 82--90, Edinburgh, Scotland, 5--8 July 1988. IEEE Computer Society.

....to calculus algebraic features allowing one in particular to deal with equality in an efficient way. In the first case, we find the works on CRS [KvOvR93] and other higher order rewriting systems [Wol93, NP98] in the second case the works on combination of calculus with term rewriting [Oka89, BT88, GBT89, JO97] to mention only a few. We come up with another point of view because of our previous works on the control of term rewriting [KKV95, Vit94, BKK98] Indeed we realized that the tool that is needed in order to control rewriting should be made explicit and could be itself naturally ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, Edinburgh (UK), pages 82--90, 1988.

.... the termination or strong normalisation of such systems and is as follows: given a terminating type system T and a terminating rewriting system R, is the combination of T and R terminating It is not surprising that this question has already received considerable attention, see for instance [1, 2, 4, 7, 11, 12, 15 17, 19, 24]. However, the situation is in our opinion not yet satisfactory, since most of the proofs of termination of a combination of a type theory and a rewriting system consist basically in redoing the proof of termination of the type theory. Ideally, one would like to have a modular proof of these ....

....because of the conversion rule. So if A = S EA R, then L(A; mix) is not necessarily contained in L(S; fi) As a consequence, we cannot immediately conclude termination of fi on L(A; mix) from termination of fi on L(S; fi) Second, a well known result originally due to Klop [21] see also [11]) states that the rewrite relation mix is not necessarily confluent on the set of pseudoterms of an algebraic type system. As a consequence, the traditional proof of subject reduction of L(A; mix) for fi breaks down [4, 9] A third problem is how to infer termination of R on some set of ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings of LICS'88, pages 82--90. IEEE Computer Society Press, 1988.

....line of inquiry involves higher order unification in the presence of equations between algebraic terms. Adding a set E of equations to the axioms for fij convertibility defines fijE equality, and determines a corresponding notion of unification, higher order E unification. Breazu Tannen showed [Bre88] that the 19 combination of algebra and typed calculus is well behaved (see also [BG89] Dou91] Bar90] a complete set of transformations for higher order E unification has been defined by Snyder in [Sny90] We have seen that by using combinators we can cast higher order unification problems ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proc. Third Annual Symp. on Logic in Computer Science, IEEE, pp. 82-90, June 1988.

....relevant both from a theoretical point of view, where one looks for general results on combination of rewriting systems, and from a practical point of view, when one develops higher order semi unification algorithms, or establishes the formal properties of algebraic functional languages. Tannen [9] showed that strong normalization and confluence are both moldular properties for the combination of algebraic TRS s with the simply typed lambda calculus equipped with fi reduction. Gallier and Tannen [10, 11] extended these results to System F. Although strong normalisation remains modular in ....

V. Breazu-Tannen. Combining algebra and higher order types. In IEEE, editor, Proceedings of the Symposium on Logic in Computer Science (LICS), pages 82--90, July 1988.

....features allowing one to deal with equality in an ecient way. In the rst case, we nd the works on CRS [ Klop et al. 1993 ] and other higher order rewriting systems [ Nipkow and Prehofer, 1998 ] in the second case the works on combination of calculus with term rewriting [ Okada, 1989; Breazu Tannen, 1988; Gallier and Breazu Tannen, 1989; Jouannaud and Okada, 1997 ] to mention only a few. We come up with another point of view because of our previous works on the control of term rewriting [ Kirchner et al. 1995; Vittek, 1994; Borovansk y 1 et al. 1998 ] Indeed we realized that the tool needed ....

V. Breazu-Tannen. Combining algebra and higherorder types. In Proceedings 3rd IEEE Symposium on Logic in Computer Science, Edinburgh (UK), pages 82-90, 1988.

....by pattern matching, provided all righthand side recursive calls are structurally smaller than the lefthand side call [12] His notion is very abstract, though, and relies on a well foundedness assumption which is satisfied in practice. Concurrently, following the pioneering work of Tannen [7, 8, 39, 9], the last two authors of the present paper proposed another solution, for a polymorphically typed calculus, based on pattern matching functional definitions following the so called General Schema [26, 27] This work was then generalized so as to cover the full Calculus of Constructions [1, 2, ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings, Third Annual Symposium on Logic in Computer Science, pages 82--90. IEEE Computer Society, 5--8 July 1988.

....righthand sides, and is easily extensible. This last property should allow us to extend the present work further by combining it with the work of Barbanera and his coauthors. Breazu Tannen was the first to advocate for combining (simply typed) calculus with (firstorder) algebraic rewriting [8]. This work had a strong influence, and developed into a whole area to which many contributed (e.g. 10, 24, 1, 2, 7] Since the present paper is by no means a survey of this subfield, we do not intend to give a full account of the literature on the subject. A partial account can be found in [22, ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings, Third Annual Symposium on Logic in Computer Science, pages 82--90. IEEE Computer Society, 5--8 July 1988. 17

....calculus has proven to be a powerful model of several aspects of modern programming languages (e.g. programmer defined functions and their parameter passing mechanisms) It would seem profitable to combine the two modes, allowing each to do what it does best. For instance, as pointed out in [Bre88], algebraic rules such as rewriting x Gamma x to 0 could be treated as code optimizations in a functional language. From the point of view of the logic of programming, the equations from which rewriting rules are defined should allow the use of first order properties of the data to be involved in ....

....a system which allows any term x Gamma x to be rewritten to 0, and a term succ(x) Gamma x to be rewritten to 1, and further suppose that terms have fixed points, so that there is a term X with X evaluating to succ(X) Then X Gamma X evaluates to 0 and to 1. The insight in Val Breazu Tannen s [Bre88] is that restriction to various type disciplines should allow lambda terms to inherit nice properties from the algebraic system. See also [BM88] Bre87] In [Bre88] it was shown that if a confluent algebraic system is added to the simply typed lambda calculus, the resulting system combining fi ....

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V. Breazu-Tannen. Combining algebra and higher-order types, in Proceedings of the Third Annual Symposium on Logic in Computer Science, pp. 82- 90, 1988.

....a more general problem, in which the types of terms contain type variables. 1 Introduction Investigation of the interaction between first order and higher order equational reasoning has emerged as an active line of research. The collective import of a recent series of papers, originating with [Bre88] and including (among others) Bar90] BG91a] BG91b] Dou92] JO91] and [Oka89] is that when various typed calculi are enriched by first order equational theories, the validity problem is well behaved, and furthermore that the respective computational approaches to verifying equations ....

....based on Huet s method; only Hustadt s is a complete unification procedure. Preliminaries We will often draw upon classical results about the lambda calculus and combinatory logic (see, for example [HS86] and use the basic results on the combination of lambda calculus and first order rewriting ([Bre88], BG91a] BG91b] Dou92] We will assume familiarity with the use of transformations on systems to study unification ( MM82] GS89a] For definitions and notations here not given explicitly, the reader is referred to [DJ90] Fix a set of equations E. Terms and equalities The types are ....

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V. Breazu-Tannen. Combining algebra and higher-order types. Proceedings of the Third Annual IEEE Symposium on Logic in Computer Science, IEEE Press, pp. 82--90, 1988.

....E must be able to prove an equation like (f (succ 0) 0) f (f (succ 0) 0) 0) if it holds in the model. Note that the equations E cannot explicitly mention f since f 62 Sigma; a proof of such an equation must involve the proof rules of equational logic. The proof of Theorem 4 uses techniques in [3, 16] and has three main lemmas. We give only an overview of the proofs of the lemmas here, which are quite technical. The lemmas use the notion of a Phi normal form: Definition 5 An algebraic context over Sigma is a context in the grammar C[ Delta] Delta] j (F 0 C[ Delta] C[ Delta] ....

....from E. Then every true equation between mixed terms in M is provable from (fi) j) and E. Proof: Sketch) As in Theorem 11, it suffices to prove that any two algebraic terms over the signature Sigma [ff g that are not provable from E are also not equal in M. Using techniques from [3] we can extract a set of Sigma equations f(s 1 = t 1 ) s n = t n )g none of which is provable from E. By the disjunctive closure of A there is a single environment ae that invalidates all the equations (s i = t i ) Using the fact that there are enough first order functions, we then ....

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Val Breazu-Tannen. Combining algebra and higher-order types. In Proceedings, Third Annual Symposium on Logic in Computer Science, pages 82--90. IEEE, 1988.

....equations of M iff E proves all equations between algebraic terms in algterms Sigma;X [ff g that are valid in M. The theorem essentially reduces the general problem to one of checking completeness on a limited fragment of the language. 4. 1 Outline of the proof We generalize techniques in [3, 19] to the case when algebraic equations are present. In broad outline, the proof has three main steps: 1. We define the set of Phi normal forms for terms over Sigma; we then prove that any term over the signature Sigma is fijE equivalent to a term in Phi normal form. 2. Suppose P and Q are ....

....axiomatized by (fi) j) and the equational theory of the corresponding algebras. Table 3: Axiomatization of (N; 0; 1; x y = y x x y = y x x (y z) x y) z x (y z) x y) z x (y z) x y x z 0 x = 0 0 x = x 1 x = x There is a theorem due to Breazu Tannen [3] that shows that for any algebraic equational theory E, the theory fijE is truth table reducible to the theory E. Theorem 7.1 (Breazu Tannen) Suppose E is a set of equations over the signature Sigma and t 1 ; t 2 2 terms Sigma;X . Then there exist sets of algebraic equations S 1 ; Sn ....

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V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings, Third Annual Symposium on Logic in Computer Science, pages 82--90. IEEE, 1988.

.... from that in [23] for the first order case via a simulation, based on a skolemization process, using a technique introduced in [21] and then generalized in [196] There is a relation to work on typed calculus, such as the theory of constructions [239] and the theory of polymorphism [188] in [136, 135, 137] it is shown how equational axioms can be consistently combined with higher order calculus. In [790, 791] order sorted algebras are extended with higher order functions for functional programming, data type specification and program transformation. The main results are a new subtype relation, a ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proc. 3rd IEEE Symp. on Logic in Computer Science, 1988.

....f(t 1 = t 2 ) 2 E j t 1 ; t 2 2 algterms S;f f i i i g;X g: Note that the theorem concerns theories, not models, which emphasizes the syntactic nature of the proof. At the end of the section, we use the theorem to deduce a fact about models. 4. 1 Outline of the proof We generalize techniques in [Breazu Tannen, 1988, Statman, 1982] to the case when algebraic equations are present. In broad outline, the proof has two main steps: 1. Let FO be a set of first order variables, with an infinite number of variables of each first order type. Let E 2 = f(t 1 = t 2 ) 2 E j t 1 ; t 2 2 algterms S;FO;X g If E 1 proves ....

....states that the equational theory of mixed terms in a model is completely determined by the structure of the first order types, is difficult to apply. We have proposed two sufficient conditions, and given a few corollaries. Our theorems can be used in combination with a theorem of Breazu Tannen [Breazu Tannen, 1988] for deducing decidability of various theories. Theorem 7.1 (Breazu Tannen) Suppose E is a set of equations over the signature S and t 1 ; t 2 2 terms S;X . Then there exist finite sets of algebraic equations S 1 ; S n , effectively computable from t 1 = t 2 , such that t 1 = bhE t 2 iff ....

Breazu-Tannen, V. (1988). Combining algebra and higher-order types. In Proceedings, Third Annual Symposium on Logic in Computer Science, pages 82--90. IEEE.

.... rewrite rules and fi reduction can be combined without loss of their useful properties (for example, strong normalization and confluence are preserved under the combination of typed LC and first order TRS) This is supported by a number of results for a broad range of type systems and calculi [12, 13, 14, 20, 23, 9], but still lacks evidence in order to be completely accepted in its full generality. More specifically, all the systems studied in the papers mentioned above have explicit type disciplines (also called a la Church) i.e. type disciplines where terms come together with types and, hence, each term ....

V. Breazu-Tannen. Combining algebra and higher-order types. In Proceedings of LICS'88, pages 82--90, 1988.

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